- Sefer B'Reshit: the Yerushalmi (Sotah 1:10) already refers to it as Sefer B'reshit. This is also found in the Zohar (Raya Mehemna Vol. II Parashat Mishpatim 119b).
- Sefer Sh'mot: The Midrash Lekah Tov (11th cent.) has a little rhyme at the end of Parashat Pekudei that refers to Sefer Sh'mot.
More common is "Sefer V'eleh Sh'mot" found in many Midrashim (although I have not seen it in critical editions). E.g. B'reshit Rabba (Vilna) (3:5, 65:8), Shir HaShirim Rabba (Vilna) (4:7), Midrash Tanhuma (Warsaw) Parashat Veyera (1). And in later (but still medieval) sources such as Midrash Aggada (Buber) B'midbar (21:14) and Yalkut Shimoni B'reshit (4). Also Rambam's Hilkhot Sefer Torah (8:7) and Ramban's introduction to Sefer Sh'mot. It is also found in the early Geonic work Seder Tannaim V'ammoraim (2:26).
- Sefer Vayikra: B'reshit Rabba (ed. Albeck) to Parashat B'reshit (3) refers to it as Sefer Vayikra. Rambam (12th cent.) does as well in Hilkhot Sefer Torah (8:9)
- Sefer B'midbar: Rashi (11th century) refers to Sefer B'midbar in Sh'mot (28:36) and Yehoshua (17:14). Rashbam (12th cent.) does as well in Bava Batra (124b), as does Rabbenu Hayyim Paltiel (13th cent.) in B'reshit (26:21).
There are also a couple of references to it as Sefer B'midbar Sinai (Rambam's Hilhkot Sefer Torah 8:9), from the 12th century, and Yad Ramah to Sanhedrin (43b) also from the 12th century.
- Sefer D'varim: R. Hayyim Paltiel (beginning of B'midbar) refers to it as Sefer D'varim. Paneah Razza (14th cent.) has a short rhyme at the end of D'varim that refers to Sefer D'varim.
Slightly more common is referring to it as Sefer "Eleh HaD'varim" (which is still distinct from the more familiar Hazalic name of Mishneh Torah". The term "Sefer Eleh Had'varim" is found in Midrash Tannaim (D'varim 33:1). And in Rambam's Hilkhot Sefer Torah (8:10) and in Ibn Ezra (12th cent.) to (Sh'mot 18:1), and later Rishonim such as Ralbag (Shoftim 10:4).
The earliest singles source that refers to all five that I have found, is R. Hayyim Paltiel (beginning of B'midbar) who writes:
וס"ת של ה' חומשי תורה מים לו, ספר בראשית במצרים, ספר שמות (ל)מסעיהם, ספר ויקרא במדבר סיני, ספר במדבר יריחו, ספר דברים לעיני כל ישראל, מלמד מי שיש בידו מים כלומר שיודע ה' חומשי תורה עיקר התורה שלו ותורה נקראת מים.
While the possibility of scribal emendations before works were printed is possible, from the large number of early sources (not mentioned above were Sh'iltot, Bahag, and later Rashi and Ibn Ezra) that refer to Sefer B'reshit, and the dearth of early sources for the other books (except to a lesser degree Vayikra), it seems that this name is probably very old. Probably even Hazalic.
As indicated above, Sh'mot and and Sefer D'varim seem the latest.